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Axiom ax-4 1288
Description: Axiom of Specialization. A quantified wff implies the wff without a quantifier (i.e. an instance, or special case, of the generalized wff). In other words if something is true for all x, it is true for any specific x (that would typically occur as a free variable in the wff substituted for φ). (A free variable is one that does not occur in the scope of a quantifier: x and y are both free in x = y, but only x is free in yx = y.) This is one of the axioms of what we call "pure" predicate calculus (ax-4 1288 through ax-7 1234 plus rule ax-gen 1235). Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77).

Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1235. Conditional forms of the converse are given by ax-12 1289, ax-15 1681, ax-16 1502, and ax-17 1297.

Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from x for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 1472.

The relationship of this axiom to other predicate logic axioms is different than in the classical case. In particular, the current proof of ax4 1809 (which derives ax-4 1288 from certain other axioms) relies on ax-3 7 and so is not valid intuitionistically. (Contributed by NM, 5-Aug-1993.)

Assertion
Ref Expression
ax-4 (xφφ)

Detailed syntax breakdown of Axiom ax-4
StepHypRef Expression
1 wph . . 3 wff φ
2 vx . . 3 set x
31, 2wal 1231 . 2 wff xφ
43, 1wi 4 1 wff (xφφ)
Colors of variables: wff set class
This axiom is referenced by:  ax-12  1289  hbequid  1293  ax5o  1304  ax5  1306  ax6o  1307  ax6  1309  a4i  1312  a4s  1313  a4sd  1314  hbnt  1315  hbim  1319  19.3  1331  19.21  1334  19.21bi  1337  hbimd  1343  19.21t  1345  a6e  1382  19.12  1390  19.38  1394  ax9o  1410  hbae  1429  drex1  1441  equveli  1456  sb2  1464  ax11  1513  ax11b  1514  dfsb2  1519  sbf3t  1545  hbsb4  1546  hbsb4t  1547  a16gb  1570  sb56  1591  sb6  1592  sbal1  1665  sbalyz  1667  mopick  1745  2eu1  1763  ax11indalem  1815  ax11inda2ALT  1816
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